In Exercises 64–70, graph each polar equation. Be sure to test for symmetry. r = 2 + 2 sin θ

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Identify the given polar equation: \(r = 2 + 2 \sin \theta\).

Recall that \(r\) represents the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis in polar coordinates.

Test for symmetry by checking the following: - Symmetry about the polar axis (x-axis): Replace \(\theta\) with \(-\theta\) and see if the equation remains unchanged. - Symmetry about the line \(\theta = \frac{\pi}{2}\) (y-axis): Replace \(\theta\) with \(\pi - \theta\). - Symmetry about the pole (origin): Replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\).

Create a table of values by choosing several values of \(\theta\) (for example, \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), etc.) and calculate the corresponding \(r\) values using the equation \(r = 2 + 2 \sin \theta\).

Plot the points \((r, \theta)\) on polar graph paper or using a graphing tool, then connect the points smoothly to visualize the curve. Observe the shape and symmetry based on your earlier tests.

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Polar Coordinates and Equations

Polar coordinates represent points in a plane using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). A polar equation expresses the radius r as a function of the angle θ, allowing the graph to be plotted by calculating r for various θ values.

Graphing Polar Equations

To graph a polar equation like r = 2 + 2 sin θ, calculate r for multiple θ values between 0 and 2π, then plot the points in polar form. Connecting these points reveals the shape, which often corresponds to known curves such as limacons or cardioids.

Symmetry in Polar Graphs

Testing for symmetry helps simplify graphing and understanding polar curves. Common symmetries include symmetry about the polar axis (θ = 0), the line θ = π/2, and the pole (origin). Checking if replacing θ with -θ, π - θ, or θ + π yields the same equation indicates these symmetries.